# gram_polynomial

gram_polynomial, a MATLAB code which evaluates the Gram polynomials.

The Gram polynomial P(n,m,x) can be evaluated at a point x by:

```        P(0,m,x) = 1
P(1,m,x) = x
P(n+1,m,x) = x * P(n,m,x) - beta(n,m) * P(n-1,m,x)
```
where beta(n,m) = (m^2-n^2)*n^s/m^2/(4*n^2-1).

The polynomials are orthogonal with respect to a discrete inner product

```  (f,g) = sum ( 1 <= i <= m ) f(x(i)) * g(x(i))
```
where
```  x(i) = - 1 + ( 2*i-1)/m, 1 <= i <= m.
```

### Languages:

gram_polynomial is available in a MATLAB version.

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### Reference:

1. Germund Dahlquist, Ake Bjorck, Numerical Methods in Scientific Computing, Volume 1, SIAM, 2008, ISBN: 978-0-898716-44-3, LC: QA297.D335 2008.

### Source Code:

Last revised on 28 June 2019.